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The sequential stabilizing spline (SSS) algorithm is a remarkable algorithm for identifying linear dynamical systems. It can guarantee the stability of an estimated model by polishing the maximum modulus roots of an equation. Therefore, the root structure of an equation plays an important role in the SSS algorithm. Although the traditional power iterative method can be applied to approximate the extremal eigenvalues which have the largest modulus, it has two limitations: (1) the extremal eigenvalues must be real numbers, and (2) the structures of the extremal eigenvalues should be known a priori. In this paper, a novel power iterative method is proposed to approximate the extremal eigenvalues. Compared with the traditional power iterative method, the method in this paper can (1) determine the types of the extremal eigenvalues without prior knowledge of the matrix; (2) approximate the true values of the extremal eigenvalues regardless of their type; (3) become a worthy addition to SSS algorithm and gradient descent algorithm. Simulation examples demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose and discuss a class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth. For an odd 2 p + 1 with a positive integer p , the problem is to construct an n × n real symmetric banded matrix with bandwidth 2 p + 1 whose m × m leading principal submatrix is a given m × m real symmetric banded matrix with bandwidth 2 p + 1 and spectrum is a given set of real numbers { λ i } i = 1 n , where the number of distinct real numbers of { λ i } i = 1 n is 2 k when m = p k , or 2 k + 1 when p k < m < p ( k + 1 ) , where m , n and k are positive integers and m < n . We point out that the well-known double dimensional (DD) problem is a special case of our proposed inverse eigenvalue problems. The necessary and sufficient condition for the solvability of the above inverse eigenvalue problem is derived, and the target real symmetric banded matrix can be constructed by the block Lanczos algorithms when the inverse eigenvalue problem is solvable. Several numerical examples show that our algorithms are feasible. Some concluding remarks are introduced. [ABSTRACT FROM AUTHOR]

The rank of an eigenvector of an unreduced real symmetric tridiagonal matrix can be determined by just knowing the signs of the elements of the eigenvector and the signs of the off-diagonal entries of the tridiagonal matrix. Surprisingly, no arithmetic operations involving real numbers are required to determine this ordinal count. The absence of real arithmetic operations guarantees an error free algorithm. Thus, it is possible to rank and order eigenvectors without knowing the corresponding eigenvalues.It is known that the singular value decomposition (SVD) of bidiagonal matrices are closely related to three tridiagonal eigenvalue problems. Using this connection, it is possible to order singular vectors of bidiagonal matrices without knowing the singular values. Again, no real arithmetic is needed.The ordinal count is a theoretical result, which is valid in exact arithmetic. If eigenvectors and singular vectors are poorly determined in floating-point arithmetic then the ordering procedure can detect faulty eigenpairs. Three standard symmetric eigenproblem routines and two SVD routines from LAPACK are investigated to verify whether the ordinal counts give valid results for computed eigenvectors. For some difficult problems such as the Wilkinson Wn+ matrix, the ordinal counts disagree with that given by the routines for n&ges;25. In general, standard LAPACK routines appear to be reasonably resilient even though the routines are not particularly designed to be robust in this context. For some classes of tridiagonals, no failures were detected for n exceeding 3000. However, one of the new LAPACK routines cannot be considered to be robust since it destroys information by thresholding small eigenvector elements to zero. An SVD routine also gave poor results for graded matrices. This could be due to a bug.Many eigenvalue algorithms do a global sort of the eigenvalues. However, if the ordinal count is reliable then ranking of eigenvectors across processors can be achieved without global communication. [Copyright &y& Elsevier]